Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

10 EMMANUEL AMIOT
be true in the smaller group, then the ‘Universal CM Complement’ can be used there, speeding up the process. This is
useful both for exhaustive catalogues of Vuza canons, and for trying to build counterexamples.
Finally, though the Universal CM Complement is only supposed to work when (T2) is satisfied, paradoxically it might
be the best way to construct counterexamples to the (T2) conjecture, as such complements B for a given SA are by
construction overloaded with superfluous cyclotomic factors; hence it may be hoped that some complements A of the
complement B will lack one or two products of elements of SA in their RA, i.e. (T2) might be false though A ⊕ B = Zn.
Moduli below 900 are unlikely to be productive in that respect, so we will need finer programs, or faster computers; 10 but
using this wealth of new ideas, this now seems well within reach.
References
[1] Amiot, E., Why Rhythmic Canons are Interesting, in: E. Lluis-Puebla, G. Mazzola et T. Noll (eds.), Perspectives of Mathematical and
Computer-Aided Music Theory, EpOs, 190–209, Universität Osnabrück (2004).
[2] Amiot, E., Some new canons, talk given in the MaMuX seminar, January 2004, online at
http://canonsrythmiques.free.fr/someNewCanons.pdf.
[3] Amiot, E., Rhythmic canons and Galois theory, Grazer Math. Ber., 347 1–25 (2005).
[4] Amiot, E., À propos des canons rythmiques, Gazette des Mathématiciens, 106 (2005).
[5] Amiot, E. A Mathematica notebook with all results and code for n = 120, http://canonsrythmiques.free.fr/nb/
[6] Agon, C., Andreatta, M. (dir.), Mosaïques et pavages en musique, collection “Musiques/Sciences”, Ed. Ircam-Delatour France, Sampzon,
2010.
[7] Andreatta, M., On group-theoretical methods applied to music: some compositional and implementational aspects, in: E. Lluis-Puebla, G.
Mazzola et T. Noll (eds.), Perspectives of Mathematical and Computer-Aided Music Theory, EpOs, 122–162, Universität Osnabrück, 2004.
[8] Andreatta, M., Gruppi di Hajós, Canoni e Composizioni., Tesi di Laurea, Facoltà di Matematica dell’Universitá degli Studi di Pavia, 1996.
[9] Andreatta, M., Méthodes algébriques en musique et musicologie du XXe siècle: aspects théoriques, analytiques, et compositionnels., thèse
de doctorat, EHESS/Ircam, 2003.
[10] Andreatta,M., Agon, C., Chemillier, M., OpenMusic et le problème de la construction des canons musicaux rythmiques, in: Proceedings
JIM 99, 3D, Issy les Moulineaux, 1999.
[11] Coven, E., and Meyerowitz, A., Tiling the integers with one finite set, in: J. Alg., 212:161-174, (1999).
[12] DeBruijn, N.G., On Number Systems, Nieuw. Arch. Wisk. (3) 4, 15–17 (1956).
[13] Fripertinger, H. Remarks on Rhythmical Canons, Grazer Math. Ber., 347, 55–68 (2005).
[14] Fripertinger, H. Tiling problems in music theory, in: E. Lluis-Puebla, G. Mazzola et T. Noll (eds.), Perspectives of Mathematical and
Computer-Aided Music Theory, EpOs, 149–164, Universität Osnabrück (2004).
[15] Gilbert, E., Polynômes cyclotomiques, canons mosaïques et rythmes k-asymétriques, mémoire de Master ATIAM, Ircam, May (2007).
[16] Fidanza, G., Canoni ritmici a mosaico, tesi di laurea, Universit degli Studi di Pisa, 2007.
[17] Hajós, G., Sur les factorisations des groupes abéliens, in: Casopsis Pest. Mat. Fys., 74:157-162 (1954).
[18] Kolountzakis, M. & Matolcsi, M., Complex Hadamard Matrices and the spectral set conjecture, http://arxiv.org/abs/math.CA/0411512.
[19] Kolountzakis, M. & Matolcsi, M., Algorithms for translational tiling, Journal of Mathematics and Music, special issue on tiling problems
in music, (2009).
[20] Laba, I., The spectral set conjecture and multiplicative properties of roots of polynomials, J. London Math. Soc. 65 , 661-671 (2002).
[21] Lagarias, J., and Wang, Y., Tiling the line with translates of one tile, in: Inv. Math., 124:341-36 (1996).
[22] Sands, A.D., The Factorization of abelian groups, Quart. J. Math. Oxford, 10(2):45-54, 1959.
[23] Sedgewick, R., Algorithms, Addison-Wesley, (2008).
[24] Szabó, S., A type of factorization of finite abelian groups, Discrete Math. 54, 121-124 (1985).
[25] Tao, T., Fuglede’s conjecture is false in 5 and higher dimensions, online at http://arxiv.org/abs/math.CO/0306134.
[26] Vuza, D.T., Supplementary Sets and Regular Complementary Unending Canons, in four parts in: Perspectives of New Music, No 29(2)
22-49; 30(1), 184-207; 30(2), 102-125; 31(1), 270-305 (1991-1992).
10 A personal computer was unable to give the aperiodic complements of a CM Universal Complement in Z900 in a fortnight.